Evaluate the Integral

Problem

(∫_0^2*π)(sin(x)*d(x))

Solution

1. Identify the antiderivative of the function sin(x)

(∫_^)(sin(x)*d(x))=−cos(x)

2. Apply the Fundamental Theorem of Calculus by evaluating the antiderivative at the upper limit 2*π and the lower limit 0

[−cos(x)](2*π)0

3. Substitute the limits of integration into the expression.

−cos(2*π)−(−cos(0))

4. Simplify the trigonometric values, noting that cos(2*π)=1 and cos(0)=1

−1−(−1)

5. Calculate the final sum.

−1+1=0

Final Answer

(∫_0^2*π)(sin(x)*d(x))=0

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